1. Area Between Two Curves
How do you find the area between two curves?

2. What is the area underneath the curve of 𝑦=5𝑥− 𝑥 2 from 𝑥=0 to 𝑥=4?
0 4 5𝑥− 𝑥 2 𝑑𝑥
= 𝑥 2 − 1 3 𝑥
=18 2 3
What is the area underneath the curve of 𝑦=𝑥 from 𝑥=0 to 𝑥=4?
0 4 𝑥 𝑑𝑥
= 𝑥 =8
What is the area between the curves of 𝑦=5𝑥− 𝑥 2 and 𝑦=𝑥 from 𝑥=0 to 𝑥=4?
0 4 5𝑥− 𝑥 2 𝑑𝑥
0 4 5𝑥− 𝑥 2 −𝑥 𝑑𝑥
0 4 4𝑥− 𝑥 2 𝑑𝑥
− 0 4 𝑥 𝑑𝑥
= −10=8 2 3

3. Note: Although Pink does not show up well on computer, does show up on projected image.− = − =

4. Area between two curves:
When 𝑓 𝑥 ≥𝑔 𝑥
for all x between 𝑎,𝑏
Use highlighter to highlight 𝑓(𝑥) graph.
𝑨= 𝒂 𝒃 𝒇 𝒙 −𝒈 𝒙 𝒅𝒙

5. Example 1
Find area of the region bounded by the graphs of 𝑦= 𝑥 2 +2 and 𝑦=−𝑥 between 𝑥=0 and 𝑥=1.
= 𝑥 2 +𝑥+2 𝑑𝑥
Don’t use class-time to solve. Just have the students set up the integral.

6. Find area between 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥.
Example 2: Region between two intersecting graphs.
Find area between 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥.
Find where the graphs intersect.
− 𝑥 2 +𝑥−2 = 0
Since 𝒇 𝒙 ≥𝒈 𝒙 between −𝟐,𝟏
−𝟐 𝟏 𝟐− 𝒙 𝟐 −𝒙 𝒅𝒙

7. −𝟐 𝟏 𝟐− 𝒙 𝟐 −𝒙 𝒅𝒙

8. If curves intersect at more than one point.

10. Horizontal Area
Sometimes it will be more convenient to find the area of a region by differentiating with respect to y instead of x.

11. -You will integrate (right function – left function).
-Setup your integral from bottom to top and use the y values instead of the x values.
−2 1
-You will integrate (right function – left function).

13. Integrate this same function using 𝑥
Integrate this same function using 𝑥. You must divide the integral into two parts.

15. Summary Non-Intersecting Graphs Intersecting Graphs
Graphs with Multiple Intersections
Finding Area Horizontally

16. Homework
Area Between Curves Section 7.1 (1-23 odd, 27, 29) (omit 11) *For odd, set up on paper but you may use your calculator to integrate.*

17. DELETED SLIDES

18. Example 2: Region between two intersecting graphs.
Find area between 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥.
Find the points where graphs intersect and integrate from the x value of the left intersection to the x value of the right intersection.